Respect is an integral part of Saint's Row. It is shown by the purple half-circle below your weapon display, in the top right of your HUD. Once it fills up (from left to right), you get a 2x or 3x or whatever multiplier indicator and it starts to fill up again.

Each time you want to do a mission, you have to "pay respect" to start it. So if you have 25x respect and begin a mission, it might go down to 24x respect. Then of course you earn respect by doing the mission, so you end up getting more. You also get respect by doing all of the mini-game / activities.

You can get a respect modifier bonus based on what you are currently wearing - this includes tattoos, clothing, jewelry, etc. So say for example you're doing a Drug Trafficking mini-game and are supposed to earn 1,000 respect points for finishing a level. If you have the right mix of these items, you can get up to a 25% bonus, meaning you earn 1,250 respect points for that exact same mission.

Wearing purple clothing gets you a big bonus, and wearing platinum jewelry does as well. Any tattoo gets you an equivalent bonus, so you might as well start out with all the tattoos you can. Then hold off on buying jewelry until you can afford a platinum item in each spot.